A model for phase balancing and velocity filtering
, by William S. Harlan

Hyperbolic reflectons and convolutional wavelets are fundamental models for seismic data
processing. Each sample of a "stacked" zero-offset section can parametrize an impulsive
hyperbolic reflection in a midpoint gather. Convolutional wavelets can model source
waveforms and near-surface filtering at the shot and geophone positions. An optimized
inversion of their combined modeling equations makes explicit any inter-dependence and
non-uniqueness in these two sets of parameters.
First, I estimate stacked traces that best model the recorded data and then find non-
impulsive wavelets to improve the fit with the data. These wavelets are used for a new
estimate of the stacked traces, and so on. Estimated stacked traces model short average
wavelets with a superposition of approximately parallel hyperbolas; estimated wavelets
adjust the phase and amplitudes of inconsistent traces, including static shifts. Deconvolution
of land data with estimated wavelets makes wavelets consistent over offset; remaining
static shifts are midpoint consistent. This phase balancing improves the resolution of
stacked data and of velocity analysis.
If precise velocity functions are not known, then many stacked traces can be inverted
simultaneously, each with a different velocity function. However, the increased number of
overlain hyperbolas can more easily model the effects of inconsistent wavelets. As a
compromise, I limit velocity functions to reasonable regions selected from a semblance
velocity analysis--a few functions cover velocities of primary and multiple reflections.
Multiple reflections are modeled separately and then subtracted from marine data.
Phase inconsistencies ought also to be recognizable over midpoint. Convolutional wavelets
can be constrained with surface-consistency. Alternatively, one can model two-dimensional
stacks as a sum of diffraction hyperbolas over midpoint, much as is done over offset. Dip
moveout makes the diffraction and normal-moveout velocities equivalent.